Formula:KLS:01.10:01

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formula in Basic hypergeometric functions

Formula:KLS:01.10:03 >>

${\displaystyle{\displaystyle{\displaystyle{}\qHyperrphis{r}{s}@@{a_{1},\ldots,a_{r}}{b_{1},\ldots,b_{s}}{q}{z}{}:=\sum\limits_{k=0}^{\infty}\frac{\left(a_{1},\ldots,a_{r};q\right)_{k}}{\left(b_{1},\ldots,b_{s};q\right)_{k}}(-1)^{(1+s-r)k}q^{(1+s-r)\binomial{k}{2}}\frac{z^{k}}{\left(q;q\right)_{k}}}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle\left(a_{1},\ldots,a_{r};q\right)_{k}:=\left(a_{1};q\right)_{k}\cdots\left(a_{r};q\right)_{k}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : basic hypergeometric (or ${\displaystyle{\displaystyle{\displaystyle q}}}$-hypergeometric) function : http://dlmf.nist.gov/17.4#E1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1

Bibliography

Equation in Section 1.10 of KLS.

URL links

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<< The q-gamma function and q-binomial coefficients

formula in Basic hypergeometric functions

Formula:KLS:01.10:03 >>